Abelian Chern-Simons theory, Stokes' theorem, and generalized connections
aa r X i v : . [ g r- q c ] N ov Abelian Chern-Simons theory, Stokes’ theorem, andgeneralized connections ✩ Hanno Sahlmann a,1 , Thomas Thiemann b a Asia Pacific Center for Theoretical Physics, Pohang (Korea)Department of Physics, Pohang University of Science and Technology, Pohang (Korea) b Institute for Theoretical Physics III, Erlangen University (Germany)Max Planck Institute for Gravitational Physics, Potsdam (Germany)Perimeter Institute for Theoretical Physics, Waterloo (Canada)
Abstract
Generalized connections and their calculus have been developed in the con-text of quantum gravity. Here we apply them to abelian Chern-Simons the-ory. We derive the expectation values of holonomies in U(1) Chern-Simonstheory using Stokes’ theorem, flux operators and generalized connections. Aframing of the holonomy loops arises in our construction, and we show how,by choosing natural framings, the resulting expectation values neverthelessdefine a functional over gauge invariant cylindrical functions.The abelian theory considered in the present article is the test case for ourmethod. It can also be applied to the non-abelian theory. Results will bereported in a companion article.
Keywords:
Abelian Chern-Simons theory, loop quantum gravity,generalized connections ✩ Preprint KA-TP-09-2010
Email addresses: [email protected] (Hanno Sahlmann), [email protected] (Thomas Thiemann) The work on this article was carried out while the author was member of the Institutefor Theoretical Physics, Karlsruhe University, Karlsruhe (Germany).
Preprint submitted to Elsevier July 10, 2018 . Introduction
One of the pillars of loop quantum gravity is a sophisticated theory of func-tions on spaces of connections with compact gauge groups. It comprisesmeasure theory [1], the definition of functional derivatives [2], and spin net-works , generalizations of Wilson loop functionals. Of particular importanceto loop quantum gravity is the definition of a (Lebesgue-like) gauge invari-ant measure on such spaces. It is a natural question whether this formalismcan also be applied to other gauge theories. This has been answered affir-matively in a number of cases, such as 2d Yang-Mills theory, or Maxwelltheory in four dimensions. With the present article we want to add U(1)Chern-Simons theory on R to this list. In particular we are interested intwo (related) questions: (i) can one make sense of the path integral for thistheory as a mathematical object related to generalized connections? and (ii)can the formalism be used to derive expectation values for the path integral?The point of the present article is to affirmatively answer both questions.We should point out that U(1) quantum Chern-Simons theory (see for ex-ample [3]) in and of itself is not too interesting. We study the abelian theoryas a test case for the underlying methods. The application we have in mindis Chern-Simons theory for a non-abelian gauge group, which is more rele-vant, both from a mathematical and from a physical perspective. In fact,there is already very interesting work on the relation between non-abelianChern-Simons theory and loop quantum gravity: see for example [4, 5, 6].We report our own results in this direction in the companion article [15].U(1) Chern-Simons theory on R is defined by the action S CS [ A ] = k π Z R A ∧ d A where A is a U (1) connection. The expectation values for a collection ofholonomies h α i along non-intersecting loops α , . . . , α n can be calculated invarious ways to be h Y i ( h α i ) n i i = exp − πik X j n j Link( α j , α j ) + 2 X j 3f natural framings in section 4, and close with a short discussion of ourresults in section 5. An appendix collects some definitions and technicalresults related to the surfaces that we use. 2. Strategy In the present section we will describe our strategy to define the CS pathintegral. Let us fix the manifold to be M = R and start by listing ouringredients.(a) We denote by A the space of generalized U(1) connections (see forexample [1]). This is a space of distributional connections. It is com-pact, Hausdorff, smooth connections are dense, and it is well suitedfor measure theory.(b) We denote by µ AL the Ashtekar-Lewandowski measure [1], a non-degenerate, uniform measure on the space A .(c) A rigorous definition of the functional derivative δ/δA has been given[8]. More precisely, b X S = Z S ǫ cab δδA c d x a ∧ d x b for any surface S gives a well defined derivation on suitably differ-entiable functionals of A . We will describe its action in more detailbelow. Here we only need to mention that it has the expected adjoint-ness properties with respect to the scalar product induced by µ AL : b X † S = − b X S . (d) Stokes’ theorem relates the contour integral of an abelian connection A around a loop α bounding a surface S (as in figure 1) to the integralof the curvature over the surface S : I α A = Z Z S d A. It holds for a very general class of surfaces (so called domains of inte-gration , see for example [10]), and in particular for the types of surfacesconsidered in the present work. Since we work on R we can assume that the bundle is trivial and work with theconnection as a one-form on the base manifold. igure 1: A simple Wilson loop α and a surface S bounded by it Our strategy is now as follows. To evaluate the expectation value of a Wilsonline, we rewrite it as a surface integral over the curvature associated to theconnection. Then we observe that δS CS δA c = k π F ab ǫ abc which we use to replace the curvature under the CS path integral with func-tional derivatives. Finally we apply integration by parts to these derivatives.These manipulations are well motivated but formal. We will see howeverthat they lead to an expression that is well defined and can be evaluated inorder to obtain the desired expectation value. Let us be more explicit andconsider the CS expectation value of the product of a holonomy h α [ A ] . = exp (cid:20)I α A (cid:21) with another functional F [ A ] of the connection. We stress that the loop α can be arbitrarily knotted. The formal manipulations just described work In fact, throughout the text we could also use the word knot in place of loop , since allthe loops are piecewise analytic and hence equivalent to polygonal loops. h h α i = Z A F [ A ] h α [ A ] exp( iS CS [ A ]) d µ AL [ A ]= Z A F [ A ] exp (cid:20)Z Z S F (cid:21) exp( iS CS [ A ]) d µ AL [ A ]= Z A F [ A ] exp " πk Z Z S −→ δδA exp( iS CS [ A ]) d µ AL [ A ]= Z A F [ A ] exp " − πk Z Z S ←− δδA exp( iS CS [ A ]) d µ AL [ A ]= h exp (cid:18) πk b X S (cid:19) F [ A ] i where S is a surface bounded by α (as in figure 1). In the last line we haveexpressed our result as the expectation value of a functional differential op-erator acting on the functional F . As we have indicated above, this operatoris a well defined derivation on cylindrical functions. Therefore, if F [ A ] is,for example, a product of holonomies, it can be easily evaluated. In thisway, we will be able to recursively calculate the desired expectation values.Let us make an important remark regarding the strategy sketched above.It addresses a question that the reader may have had while reading ourdescription of the surface S bounded by the loop α : Could it not happenthat α does not bound any surface? What then? Indeed, by definition,if α is a non-trivial cycle of the manifold M , it is not the boundary of asurface. This is why we restrict ourselves to R for the present paper. Whatour approach can say in the case of not simply connected manifolds is aninteresting question that may be investigated elsewhere.Still, even for the case of M = R it may not be immediately obvious thatfor any loop, one can find a surface bounded by it. This however is assured:there are always such surfaces. One type of surface having a given loop (oreven link) as its boundary is called Seifert surface of the knot or link. Bydefinition a Seifert surface is embedded, connected, and orientable, the latterof which is important for our purposes since it ensures that the derivation X is well defined. A theorem by Pontrjagin and Frankl asserts that thereexists a Seifert surface for any knot or link. (An elegant construction ofsuch a surface is due to Seifert, hence the name.) Two examples of Seifertsurfaces are depicted in figure 2. Besides Seifert surfaces, we will also usethe notion of a compressing disc for a given loop. A compressing disc for6 igure 2: Examples for Seifert surfaces of knots and links: For the trefoil knot (right),and the Hopf rings (left) (graphics created with SeifertView [12]) a loop is a surface bounded by the loop, which is an immersion of a disc.More detailed descriptions of the surfaces used in the following, as well assome simple technical results, are compiled in the appendix.Now that we have ensured that to a given knot or link there are surfacesbounding it, a reasonable question is: Aren’t there too many? We will seethat the non-uniqueness in assigning a surface to a loop shows up in the endresult as a choice of framing for the loop. A surface bounded by the loopinduces a framing on the loop , and it is this framing that determines theself-linking in (1).With these remarks in place, we can now move towards the actual calcula-tion. Pick any vector field on the loop that is everywhere non-zero, and transversal to thesurface. Any such choice will lead to a framing, and all the framings obtained this wayare equivalent. . Calculation of expectation values In this section we will obtain the expectation value (1) using language andsome techniques from loop quantum gravity. For some technical backgroundon the surfaces used we refer to the appendix. We will work with the action S CS [ A ] = k π Z R A ∧ d A = k π Z R ǫ abc A a ∂ b A c d x and A is a real field. Then δS CS δA c = k π F ab ǫ abc with F ab the components of curvature F = dA = ∂ a A b d x a ∧ d x b , hence ǫ cab δS CS δA c d x a ∧ d x b = k π F. (2)The integrated functional derivative on the left hand side is a well knownobject in loop quantum gravity. It can be applied to holonomies, or, moregeneral, cylindrical functions and acts as a derivation b X S : b X S = Z S ǫ cab δδA c d x a ∧ d x b . For the group U (1), an integer n and a loop α , b X S h nα = inI ( S, α ) h nα (3)where I ( S, α ) is the signed intersection number between S and α (see ap-pendix Appendix A). We also note that I ( S, α ) = Link( ∂S, α ) . For a proof see the appendix. We rewrite the right hand side of (2) usingStokes’ theorem, and obtain (formally):2 πk b X S e iS CS = ie iS CS Z ∂S A ≡ iA ∂S e iS CS Thus it is useful to introduce the operator A α with commutation relations[ b X S , A α ] = I ( S, α ) . (4)8trictly speaking A α = − i ln h α is not well defined since 0 is in the discretespectrum of h α . But we will not be concerned by this here, since some of themanipulations under the path integral are formal anyway. The commutator(4) is zero when α = ∂S , since I ( S, α ) as we have defined it is zero in thiscase. This is the regularization chosen in loop quantum gravity, but one canmake other choices, and we will do so, here. A surface S endows its boundaryloop ∂S with a framing in a natural way. Just choose a smooth vector fieldon ∂S that is transversal to ∂S and nowhere tangent to the surface. Differentvector fields chosen this way are equivalent as framings, as one can easily see.Pick one of these vector-fields, v , and use it to “transport” the boundaryloop outwards: If ( ∂S )( t ) is some parametrization of the boundary loop,then ( ∂S ) ǫ ( t ) := ( ∂S )( t ) + ǫv ( t ). Let us then define[ b X S , A ∂S ] = lim t → [ b X S , A ∂S ǫ ] . (5)Let us also modify our definition of the signed intersection number I suchthat with the definition (5), the commutation relations between the b X andthe connection can still be written in the form (3). We also note that (5)leads back to the standard regularization (in which the commutator (5)vanishes) for surfaces without self-intersections.Now we can calculate the the expectation values we are interested in. Let α . . . α N be loops, S . . . S N surfaces with ∂S i = α i , and n . . . n N integers.Then formally h Y i h n i α i i = Z d µ AL [ A ] h n N α N . . . h n α " ∞ X l =0 l ! ( in A α ) l e iS CS = Z d µ AL [ A ] h n N α N . . . h n α " ∞ X l =1 l ! ( in A α ) l − πn k b X S e iS CS = Z d µ AL [ A ] h n N α N . . . h n α " − πn k ∞ X l =1 l ! ( in A α ) l − ←−− b X S ) e iS CS where in the last step we have used that b X S is anti-symmetric with respectto the measure µ AL . As a consequence the derivation now acts on somethingthat is allowed to act, and the next steps commute it through the holonomyoperators. From (4),(3) − πn k ( in A α ) l − ←−− b X S = − πn k (cid:20) ( l − inI ( S , ∂S )( inA α ) l − + ←−− b X S ( inA α ) l − (cid:21) . (6)9ith the result that we have moved the derivation past all the connectionterms coming from the same loop α . Before moving the derivation furtherleft, we want to repeat the procedure of exchanging an A α for a b X S andcommuting it left through all the A α ’s, until all of the A α ’s have beeneliminated this way. To this end we will use the recursion formula implicitin (6): It says that under the manipulations described( in A α ) l =: F ( l ) = ( k − XF ( l − 2) + EF ( l − 1) (7)where we have introduced the shortcuts X = − πn ik I ( S , ∂S ) E = − πk ←−− b X S . Let us also make the definition F (0) = 1. Note that with the understandingthat the operators b X S always stand to the left, we can use (7) as if E is a number. (7) is not easily solved explicitly, so we will work with itsexponential generating function G ( w ) := ∞ X l =0 l ! F ( l ) w l . We note that what we are really interested in is G (1) = ∞ X l =0 l ! F ( l ) . From (7) one can derive that G ( w ) must satisfy a differential equation: G ′ ( w ) = XwG ( w ) + EG ( w )The initial condition is G (0) = F (0) = 1. We find the solution G ( w ) = exp (cid:20) wE + w X (cid:21) . By evaluating at w = 1 we find that under the path integral, and using themanipulations we have described above, ∞ X l =0 l ! ( in A α ) l = e − πk ←−− b X S e − πn ik I ( S ,∂S ) . b X S further to the left, we end up with h Y i h n i α i i = exp − πn ik I ( S , ∂S ) − πk n X j =2 I ( S , α j ) Z d µ AL [ A ] h n N α N . . . h n α e iS CS . We can now repeat this procedure with the other holonomies, and, as a finalstep replace signed intersection numbers I by Gauß linking (or, in the caseof I ( S, ∂S ), self-linking ) and obtain the well known result h Y i h n i α i i = exp − πik X j n j Link( α j , α j ) + 2 X j 4. Natural framings Obviously, the result (8) depends on the surfaces S i chosen in the processof calculation. These enter through the self-linking for the loops α i that11hey define. The formalism of loop quantum gravity is using loops (or moregenerally, graphs) that are not framed. So for (8) to serve as a definition ofa functional on Cyl, one will have to make a choice of surface, and hence offraming, for each loop. One way to proceed is to leave the framing unspeci-fied and go over to the formalism of framed spin-networks [7]. But there isalso another option: One can search for ways in which loops get assignedsurfaces (and hence framings) based on their properties. Such a choice wouldthen be part of the definition of the path integral. A minimal requirementon such a choice is that it makes the expectation values (8) invariant underdiffeomorphisms, i.e., h Y i h n i α i i = h Y i h n i φ ( α i ) i for any diffeomorphism φ of R , which can be connected to the identity. Anecessary and sufficient condition in terms of a map α S α from loops tosurfaces is thus I ( S α , α ) = I ( S φ ( α ) , φ ( α ))for all loops α and all diffeomorphisms φ connected to the identity. Such amap then endows each loop with a framing such thatLink( α, α ) = Link( φ ( α ) , φ ( α )) (9)for all loops α . Let us mention two examples for such assignments of surfacesand hence framings:1. For a loop α choose S α to be a minimal compressing disc, i.e. one thatminimizes the number of intersections of the loop with the surface.The minimal number of such intersections is an invariant of the loop,the knottedness K( α ) [13], and Link( α, α ) = K( α ).2. For a loop α choose S α to be a Seifert surface. A Seifert surface doesnot self-intersect and hence Link( α, α ) = 0.We call the framing obtained from these and similar prescriptions univer-sal and natural , the former because they encompass all loops, the latterbecause of their covariance (9). (The second prescription also yields whatis called natural framing in the mathematical literature.) Making one ofthese choices, (8) becomes a function of unframed loops. Still it is not afunctional on Cyl since not all functionals in Cyl are linear combinations ofmultiloops. It is however well known that the gauge invariant functionalscan all be written as such linear combinations and on those (8) defines awell defined functional. 12e remark that for graphs that can be decomposed into loops, the aboveprescriptions will also give a notion of ‘framed graph’, and hence also anotion of framing for gauge invariant functionals in Cyl. It would be inter-esting to compare this in detail to the notion of framed spin network of [7].We suspect that the notions will turn out to be the same. 5. Closing remarks In the present article we have done two things. On the one hand we haveexplained how the Wilson loop expectation value (1) of U(1) Chern-Simonstheory can be interpreted as a functional over gauge invariant cylindricalfunctions by using a universal framing prescription, and we have given twoexamples of such a prescription. On the other hand we have derived the ex-pectation values (1) using some heuristic manipulations of the path integralof the theory. These manipulations used the fact that certain functionalderivatives have well defined action on Wilson loops in the form of fluxoperators , well known from loop quantum gravity.Since U(1) Chern-Simons theory is not very interesting in itself, the poten-tial value of the work presented above lies in the methods used. Using auniversal framing prescription to describe a theory that needs framing withthe mathematical methods based on generalized connections may be usefulfor different theories. And we have used a regularization of the (abelian)flux that differs from the one used in LQG. This may also find applicationselsewhere. But, most importantly, the technique we have used to computethe expectation values (1) seem to be applicable also to Chern-Simons the-ory with compact, non -abelian gauge group. In particular, in [15] we havestarted to investigate the case of SU(2), relevant to knot theory [14], Eu-clidean gravity in three dimensions, and maybe even to four dimensionalgravity via the Kodama state. Our findings show that again flux opera-tors can be defined that replace holonomies under the path integral, using anon-abelian generalization of Stokes’ theorem. The regularization of theseoperators is however much more complicated than in the abelian case. Inparticular, since the functional derivatives do not commute for the non-abelian group, there is an ordering ambiguity. In fact, something like it isexpected since it may be the technical reason for the occurrence of quan-tum SU(2) in the quantum theory. In [15], we show that a certain naturalmathematical structure in the theory of Lie, algebras – the so-called Duflo ap – can be used to perform the ordering, and we calculate the results forsome simple cases. Acknowledgements HS gratefully acknowledges funding for the earlier stages of this work througha Marie Curie Fellowship of the European Union. His later research was par-tially supported by the Spanish MICINN project No. FIS2008-06078-C03-03. Appendix A. Surfaces, Gauß linking, and intersection numbers Before we start the calculation of expectation values in CS theory, we willcollect here some definitions and mathematical facts needed in the followingsections.For the moment a loop will be a smooth, compact closed one dimensionalsubmanifold in R . Also we assume loops to be tame, i.e. equivalent toa polygonal knot. Later we will have to make some adjustments to thisdefinition owing to the fact that loops in the formalism of loop quantumgravity that we are going to use are actually piecewise analytic.Let us pick an orientation of R and stick with it throughout. All the surfaceswe consider will be orientable, and we assume them to be oriented, even ifwe do not state this explicitly each time. We will also take the loops to beoriented.Given a loop α in R we consider surfaces S such that ∂S = α . We willalways assume that S and α are oriented consistently. We will use twoclasses of such surfaces, Seifert surfaces, and compressing discs. Let us givea brief description of each. Definition Appendix A.1. A Seifert surface for a loop α is an orientable,connected submanifold S such that ∂S = α . Seifert surfaces exist for any loop in R . In fact there exist a simple algorithmto construct one for a given loop. There is however no uniqueness, not even That means, if T is a positively oriented tangent vector to the loop α at a point p ,and N is the outward normal vector (tangent to S ) of S at p , then ( N, T ) is a positivelyoriented basis of T p S . 14n a topological sense: A loop has many different Seifert surfaces. For theseand other general results on Seifert surfaces see for example [11].For the definition of a compressing disc we follow [13]. That reference workswith loops in S instead of R but for the few things we need from there,the compactification makes no difference. Definition Appendix A.2. We define a compressing disc of a loop α tobe a map f from the two dimensional disc D into R such that f | ∂D = α and f | int D is transverse to α . Then f (int D ) has only finitely many intersections with α , and one canshow that for a given loop there is a minimal number of intersections thatcan be achieved by varying the compressing disc. This number is calledknottedness and is an invariant of the loop. It is shown in [13] that startingfrom a compressing disc with the image of which has n intersections withthe loop, one can always find a compressing disc with n or less intersections,which in addition is an immersion of D into R . In the following (and in themain text) we will always assume all the compressing discs to be immersions.Let us call an orientable, connected, two-dimensional submanifold of R asurface of type I , and an immersion of D into R a surface of type II . Anotion we need for both types of surfaces is that of the signed intersectionnumber between a surface and a loop. Definition Appendix A.3. For a surface S of type I and a loop α , thesigned intersection number is I ( S, α ) = X p ∈ S ∩ α κ ( p ) where κ ( p ) = +1 if the intersection is transversal and the orientations of α and S together give the orientation of the one chosen on R , κ ( p ) = − ifthe intersection is transversal and the orientations of α and S together givethe orientation opposite of the one R , and zero otherwise. For a surface of type II the definition is almost the same, except that inter-sections with a loop at a point of self-intersection of the surface may countmultiple times. Definition Appendix A.4. Let f : D → R be a surface of type II. Thenwe can find an open cover { U J } of D such that f ( U J ) is a surface of type We mean: If T is a positively oriented tangent vector to α at the intersection point p , and ( v , v ) is a positively oriented basis of T p S , then ( T, v , v ) is positively orientedin T p R . igure A.3: The types of crossings used in the calculation of the linking number of twoloops.Figure A.4: The shrinking procedure of the loop used in the proof of prop. Appendix A.5.Orientation of D is out of this page, towards the reader, intersections with positivesignature are indicated by a ‘+’, those with negative signature by a dot. I for every J . Then given a loop α we define I ( S, α ) = X J I ( f ( U J ) , α ) . It is easily checked that this definition is independent of the open cover.Another notion that we will need is that of the Gauß-linking number (orsimply linking number). It is a property of a pair of oriented loops. One ofseveral equivalent ways of defining it is the following: Given two orientedloops α , α , draw a diagram of the pair of loops. To each crossing c in thediagram, determine the quantity s c by comparison with figure A.3. Thenthe linking number is Link( α , α ) = X c s c . It is easy to see that the linking number is independent of the diagramchosen, and that it is, in fact, a topological invariant of the loops.The linking number is related to the signed intersection number as follows:16 roposition Appendix A.5. Let S be a surface of type I or II, and α bea loop. Then I ( S, α ) = Link( ∂S, α ) . Proof. For a surface of type I, the proof of the proposition is containedin the paper [9] as appendix 1, and so we won’t reproduce it here. Forsurfaces of type II the argument is as follows: Let a type II surface S anda loop α be given. Thus we have an immersion f of D into R , the imagebeing S . Let us describe the situation by depicting D , together with thepreimage of the intersection points of S with α . Thus we get a picture likefigure A.4(a), where we have additionally kept track of the signature κ ofthe intersection. The signed intersection number can be read of from thediagram, by subtracting the number of dots (preimages of negative signatureintersections) from the number of pluses (preimages of positive signatureintersections). Now we will construct a family of new type II surfaces S ( t )from S , by suitably restricting the domain of the immersion f , or, in otherwords by shrinking the boundary loop. To this end, we consider a homotopyof loops β ( t ) in D , with β (0) = ∂D and β (1) a loop that has no preimagesof intersections within the disc that it bounds (such a loop is depicted inA.4(d)), and such that its image under f has no linking with α . Obviouslythe latter two conditions are compatible, and many such loops exist. Wedefine the surface S ( t ) as the image under f of the disc bounded by β ( t ).Since f is only an immersion, f ( β ( t )) for any given t = 0 is not necessarilya loop: It can have self-intersections. One can convince oneself, however,that one can choose the homotopy in such a way that the self-intersectionsof f ( β ( t )), if present at all, are isolated points, and such that f ( β (1)) doesnot have any self-intersections. We will consider such a homotopy in thefollowing. Let us keep track of Link( α, β ( t )) and I ( S ( t ) , α ) as we vary t Obviously we start withLink( α, f ( β (0))) = Link( α, ∂S ) I ( S (0) , α ) = I ( S, α ) . As long as f ( β ( t )) does not develop self-intersections or β ( t ) moves past thepre-image of an intersection of S with α , nothing changes. When f ( β ( t ))develops a self-intersection, Link( α, β ( t )) is a priori not well defined. But wewill define it to be whatever one gets when one removes the self-intersectionsby slightly moving f ( β ( t )) such that the self-intersections disappear, withouthowever moving f ( β ( t )) through α . The result is independent of the preciseway this done, since Link can be computed using only the crossings of α with f ( β ( t )), no self-crossings of f ( β ( t )). I ( S ( t ) , α ) is obviously insensitive17o self-intersections of f ( β ( t )). If on the other hand β ( t ) moves past thepre-image of an intersection of S with α (as depicted in figures A.4(b),(c)),both quantities change: For a positive signature intersection, I ( S ( t after ) , α ) = I ( S ( t before ) , α ) − f ( β ( t after )) , α ) = Link( f ( β ( t before )) , α ) − . Similarly, for passing a negative signature intersection, I ( S ( t after ) , α ) = I ( S ( t before ) , α ) + 1Link( f ( β ( t after )) , α ) = Link( f ( β ( t before )) , α ) + 1 . At the end of the process,Link( α, f ( β (1))) = 0 , I ( S (1) , α ) = 0 . Thus Link( α, β ( t )) and I ( S ( t ) , α ) are the same in the end, and they changein step, thus they have been equal at the beginning as well, which provesthe proposition.A final remark of this appendix concerns the differentiability category usedfor the loops and surfaces. In the description above all surfaces (and im-plicitly also their boundaries) are smooth. In loop quantum gravity thesurfaces for which the flux operators are well defined are however real an-alytic (or piecewise real analytic, defined in a suitable sense), and so arethe loops. This is to ensure that there are only finitely many transversalintersections for all pairs of compact loops and surfaces, which is in turnnecessary to make the analog, for non-abelian groups, of the commutator(3) well defined. This is however not a concern for the work presented here,since due to the abelian nature of U(1), flux operators are well defined evenon loops that intersect the underlying surface infinitely often as long as theGauss linking between the loop and the boundary of the surface is finite.This is the case if we work with piecewise analytic loops and boundaries.Thus the remaining question is whether the existence of Seifert surfaces andcompressing discs, as well as the results of this appendix continue to holdfor piecewise analytic loops. It is easy to see that Seifert surfaces continueto exist in this case, and the notion of a compressing disc can be triviallygeneralized to allow for a piecewise analytic boundary. Finally it is easy tosee that all properties remain intact. Thus, in the main text, we will alwaysassume loops to be piecewise analytic and surfaces to be smooth.18 eferences [1] A. Ashtekar and J. 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